Question

Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any

quadrilateral is cyclic
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Answer 1

Given a quadrilateral ABCD with internal angle bisectors AF, BH, CH

and DF of angles A, B, C and D respectively and the points E, F, G

and H form a quadrilateral EFGH.

To prove that EFGH is a cyclic quadrilateral.

∠HEF = ∠AEB ⠀⠀⠀ [Since,Vertically opposite angles] -------- (1)

Consider triangle AEB,

∠AEB + 1/2 ∠A + 1/2 ∠B = 180°

∠AEB = 180° – 1/2 (∠A + ∠ B) -------- (2)

From (1) and (2),

∠HEF = 180° – 1/2 (∠A + ∠ B) --------- (3)

Similarly, ∠HGF = 180°– 1/2(∠C + ∠ D) ------- (4)

From 3 and 4,

∠HEF +∠HGF = 360°– 1/2∠A +∠B +∠C +∠ D)

= 360° – 1/2 (360°)

= 360° – 180°

= 180°

So, EFGH is a cyclic quadrilateral since the sum of the opposite

angles of the quadrilateral is 180°.

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~hope it helps❣️

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Answer 2

Refers to the attachment

Hope it helps u